A modern big wave of machine learning has propagated to fluid dynamics community.
In particular, neural networks, which have a great potential as an universal approximator [1]}, [2]}, [3]}, [4]}, have acquired strong attentions from fluid mechanicians for various extensions [5]}.
Fundamental studies for closure modeling in large-eddy simulation (LES) and Reynolds Averaged Navier–Stokes (RANS) simulation can be regarded as one of enthusiastic topics in neural networks and fluid dynamics [6]}.
Gamahara and Hattori [7]} applied a multi-layer perceptron (MLP) with only one hidden layer to LES closure and compared its ability to conventional models, considering a turbulent channel flow.
An extension of a similar idea was performed by Maulik and San [8]} with a deconvolution approach.
Following these pieces of seminal work, various studies have tackled to neural network-based LES modeling so as to establish the universal closure that can be applied to a wide range of flows [9]}, [10]}, [11]}, [12]}.
For RANS modeling, a notable work here is that of Ling et al. [13]}.
They proposed a tensor-basis neural network (TBNN), which can guarantee a Galilean invariance, and tested the model for flows in a duct and over a wavy-wall.
The TBNN have been extended to various flow configurations [14]} and more practical issues, e.g., uncertainty quantification [15]}.
In addition to the aforementioned supervised methods, Novati et al. [16]} have recently proposed a reinforcement learning-based closure by considering a homogeneous isotropic turbulent flow, which enables us to expect new methods of machine learning-based turbulence modeling.
